We first present in Fig. REF  the possibility to address the \(\Delta a_\mu \)  anomaly with dark U(1) gauge boson and the \(Z_3\)  complex scalar model,
where the \((g-2)_e\)  exclude a lot of parameter spaces of light gauge or scalar masses, see pink regions. Generally, the \((g-2)_e\)  bound can be avoided when the dark gauge boson (or \(Z_3\)  scalar) couple with the \(e\)  and \(\mu \)  with a ratio [1]}: \(r_V=\sqrt{g_D^e/g_D^\mu }\) , (or \(r_S=\sqrt{y_{ee}/y_{\mu \mu }}\) ). The NA64 would exclude more parameter spaces when invisible decay of dark gauge boson is taken into account [2]}, [3]}. The GWs from FOPT for different benchmarks for dark U(1) and \(Z_3\)  scalar are marked in the two plots.
<TABLE><FIGURE><FIGURE><TABLE>